📚 Distributions in IB Mathematics | IB数学中的分布
In probability and statistics, a distribution describes how the probabilities of a random variable are spread over its possible values. Understanding different types of distributions — both discrete and continuous — is essential for IB Mathematics, whether you are following the Analysis and Approaches (AA) or Applications and Interpretation (AI) route. This article systematically covers the key distributions you will encounter, including binomial, Poisson, geometric, and normal distributions, together with their properties, formulas, and exam applications. We will also discuss approximations and common pitfalls.
在概率与统计中，分布描述了一个随机变量的概率如何在其可能取值上分布。无论你学习的是分析与方法（AA）课程还是应用与解释（AI）课程，理解不同类型的分布——离散型和连续型——对于 IB 数学都至关重要。本文系统涵盖你将遇到的关键分布，包括二项分布、泊松分布、几何分布和正态分布，以及它们的性质、公式和考试应用。我们还会讨论近似计算和常见陷阱。
1. Random Variables and Probability Distributions | 随机变量与概率分布
A random variable is a variable whose numerical value is determined by the outcome of a random experiment. Random variables are classified as discrete if they take a countable number of distinct values, for instance the number of defective items in a batch. Continuous random variables, on the other hand, can take any value within a given interval, such as the height of a student.
随机变量是其数值由随机试验结果决定的变量。如果随机变量取可数个不同的值，比如一批产品中的缺陷品数量，则被分类为离散型随机变量。而连续型随机变量可以在给定区间内取任何值，例如学生的身高。
For a discrete random variable X, the probability distribution is a list of all possible values x together with their associated probabilities P(X = x). Two fundamental rules apply: 0 ≤ P(X = x) ≤ 1 for every x, and the sum of all probabilities equals 1. This distribution is often displayed in a table or a bar chart.
对于离散随机变量 X，概率分布是一个包含所有可能取值 x 及其相应概率 P(X = x) 的列表。两条基本规则适用：对所有 x，0 ≤ P(X = x) ≤ 1，且所有概率之和为 1。该分布通常用表格或条形图表示。
Continuous random variables require a probability density function (PDF), denoted f(x). The probability that X lies in the interval [a, b] is the area under the density curve: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. The total area under the PDF over all real numbers is 1.
连续随机变量需要概率密度函数 f(x)。X 落在区间 [a, b] 内的概率是密度曲线下的面积：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。PDF 在所有实数上的总面积为 1。
2. Discrete Uniform Distribution | 离散均匀分布
The simplest discrete distribution is the discrete uniform distribution, where all n possible outcomes are equally likely. If X takes the values x₁, x₂, …, xₙ, then P(X = xᵢ) = 1/n for each i. Rolling a fair six‑sided die is a classic example.
最简单的离散分布是离散均匀分布，其中所有 n 个可能结果等可能发生。如果 X 取值 x₁, x₂, …, xₙ，则有 P(X = xᵢ) = 1/n。掷一个公平六面骰子就是一个经典例子。
When the outcomes are consecutive integers from 1 to n, the expected value and variance have neat closed forms: E(X) = (n + 1)/2 and Var(X) = (n² − 1)/12. These formulas are particularly useful for quick calculations without needing to sum many terms.
当结果为从1到 n 的连续整数时，期望值和方差有简洁的闭合形式：E(X) = (n + 1)/2，Var(X) = (n² − 1)/12。这些公式在无需对多项求和时特别有用。
3. Binomial Distribution: Conditions and Formula | 二项分布：条件与公式
A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. Four conditions must be met: the number of trials n is fixed in advance; each trial has only two possible outcomes (success or failure); the trials are independent; and the probability of success p remains constant across all trials.
二项分布用于模拟在固定次数的独立伯努利试验中成功的次数。必须满足四个条件：试验次数 n 事先固定；每次试验只有两种可能结果（成功或失败）；各次试验相互独立；且每次试验的成功概率 p 保持恒定。
If X follows a binomial distribution, written X ~ B(n, p), then the probability of obtaining exactly r successes is given by P(X = r) = C(n, r) pʳ qⁿ⁻ʳ, where r = 0, 1, …, n and q = 1 − p. The binomial coefficient C(n, r) counts the number of ways to choose r successes out of n trials.
如果 X 服从二项分布，记为 X ~ B(n, p)，则得到恰好 r 次成功的概率为 P(X = r) = C(n, r) pʳ qⁿ⁻ʳ，其中 r = 0, 1, …, n，q = 1 − p。二项式系数 C(n, r) 计算从 n 次试验中选出 r 次成功的方法数。
You can evaluate binomial probabilities using your GDC’s binomial PDF (for a single r) or binomial CDF (for cumulative probabilities P(X ≤ r)). Always check that the conditions are valid before applying the binomial model.
你可以使用计算器的二项分布概率密度函数（用于单个 r）或二项累积分布函数（用于累积概率 P(X ≤ r)）来求值。在应用二项模型之前，一定要确保条件成立。
4. Mean and Variance of Binomial Distribution | 二项分布的均值与方差
The mean or expected value of a binomial random variable is E(X) = np. This is intuitive: if you perform n trials each with success probability p, you expect np successes on average. The variance is Var(X) = npq = np(1 − p), and the standard deviation is σ = √(npq).
二项随机变量的均值或期望值为 E(X) = np。这很直观：如果你进行 n 次试验，每次成功概率为 p，则平均期望 np 次成功。方差为 Var(X) = npq = np(1 − p)，标准差为 σ = √(npq)。
These measures of central tendency and spread are provided in the IB formula booklet, but you must be able to interpret them in context. For example, if X ~ B(20, 0.4), then the expected number of successes is 8, and the variance is 4.8, indicating a moderately spread distribution.
这些集中趋势和离散程度的度量值在 IB 公式手册中提供，但你必须在具体情境中会解读它们。例如，若 X ~ B(20, 0.4)，则期望成功次数为 8，方差为 4.8，表明分布中等程度分散。
5. Poisson Distribution | 泊松分布
The Poisson distribution is used to model the number of events that occur in a fixed interval of time or space, provided the events happen independently and at a constant average rate λ. Typical examples include the number of phone calls received by a call centre per minute, or the number of flaws per metre of fabric. This distribution is mainly studied in the AI HL course.
泊松分布用于模拟固定时间或空间间隔内事件发生的次数，前提是事件独立发生且以恒定的平均率 λ 发生。典型的例子包括呼叫中心每分钟接到的电话数，或每米布料上的瑕疵数。该分布主要在 AI HL 课程中学习。
If X ~ Po(λ), then P(X = r) = (e⁻λ λʳ) / r! for r = 0, 1, 2, …. Unlike the binomial distribution, there is no fixed upper limit for r, and the mean and variance are both equal to λ: E(X) = λ, Var(X) = λ. This equal mean–variance property is a key characteristic used to identify a Poisson model.
若 X ~ Po(λ)，则 P(X = r) = (e⁻λ λʳ) / r!，r = 0, 1, 2, …。与二项分布不同，r 没有固定的上限，且均值和方差都等于 λ：E(X) = λ，Var(X) = λ。这种均值等于方差的性质是识别泊松模型的关键特征。
6. Geometric Distribution | 几何分布
The geometric distribution focuses on the number of trials needed to obtain the first success in a sequence of independent Bernoulli trials. If the probability of success is p, and X is the trial number on which the first success occurs, then X ~ Geo(p). The probability that the first success happens on the r‑th trial is P(X = r) = qʳ⁻¹ p for r = 1, 2, 3, …, where q = 1 − p.
几何分布关注在一系列独立伯努利试验中获得首次成功所需的试验次数。如果成功概率为 p，且 X 是首次成功发生的试验序号，则 X ~ Geo(p)。首次成功发生在第 r 次试验的概率为 P(X = r) = qʳ⁻¹ p，其中 r = 1, 2, 3, …，q = 1 − p。
The expected value is E(X) = 1/p, and the variance is Var(X) = q / p² = (1−p) / p². For instance, if you are throwing darts and hit the bullseye with probability 0.2, you would expect to need 5 throws on average to hit it for the first time. This distribution is included in the AI HL syllabus.
期望值为 E(X) = 1/p，方差为 Var(X) = q / p² = (1−p) / p²。例如，如果你投掷飞镖且击中靶心的概率为 0.2，则平均需要 5 次投掷才能首次击中。此分布包含在 AI HL 课程大纲中。
7. Continuous Random Variables and Probability Density Functions | 连续随机变量与概率密度函数
For continuous random variables, the probability of the variable taking any single exact value is zero; probabilities are only defined over intervals. The PDF f(x) must satisfy f(x) ≥ 0 for all x, and the total area under the curve must equal 1: ∫₋∞⁺∞ f(x) dx = 1. The probability that X lies between a and b is P(a < X < b) = ∫ₐᵇ f(x) dx.
对于连续随机变量，变量取某个精确单值的概率为零；概率仅在区间上定义。概率密度函数 f(x) 必须满足对所有 x 有 f(x) ≥ 0，且曲线下的总面积必须为 1：∫₋∞⁺∞ f(x) dx = 1。X 介于 a 和 b 之间的概率为 P(a < X < b) = ∫ₐᵇ f(x) dx。
The cumulative distribution function F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt gives the probability that the random variable is at most x. The median m satisfies F(m) = 0.5, and quartiles can be found similarly. In IB exams, you may be given a PDF and asked to find unknown constants, probabilities, or the median.
累积分布函数 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt 给出随机变量至多为 x 的概率。中位数 m 满足 F(m) = 0.5，四分位数也可类似求得。在 IB 考试中，你可能会遇到给定 PDF 求未知常数、概率或中位数的问题。
8. The Normal Distribution | 正态分布
The normal distribution is the most important continuous distribution in statistics. It is symmetric and bell‑shaped, completely characterised by its mean μ and variance σ². We write X ~ N(μ, σ²). Its probability density function is f(x) = 1/(σ√(2π)) · e^(−(x−μ)²/(2σ²)).
正态分布是统计学中最重要的连续分布。它是对称的钟形曲线，完全由其均值 μ 和方差 σ² 刻画。我们记为 X ~ N(μ, σ²)。其概率密度函数为 f(x) = 1/(σ√(2π)) · e^(−(x−μ)²/(2σ²))。
The empirical rule (68–95–99.7 rule) states that approximately 68% of observations fall within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ. This rule helps estimate probabilities rapidly without a calculator.
经验法则（68–95–99.7 法则）指出，大约 68% 的观测值落在 μ ± σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。这个法则有助于快速估算概率，而无需使用计算器。
To find probabilities like P(X < a), use the normalcdf function on your GDC, entering the lower bound (often −10⁹⁹ for negative infinity), upper bound, mean, and standard deviation. For inverse problems, use invNorm with the cumulative probability, mean, and standard deviation.
要计算像 P(X < a) 这样的概率，可以使用计算器上的 normalcdf 功能，输入下界（常用 −10⁹⁹ 表示负无穷）、上界、均值和标准差。对于逆问题，使用 invNorm 并输入累积概率、均值和标准差。
9. Standard Normal
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